They left off some qualifiers. According to wikipedia, it's applicable to a continuous function mapping a nonempty compact convex set to itself. Which ends up making a lot more sense to me.
Shaking has to be continuous, the particles move quickly and erratically, but they trace continuous paths.
In an ideal system with points instead of particles, shaking would be continuous.
Think of the 1D variant. If you shuffle a deck of cards, but require that to be ‘continuous’, few shuffles remain (I think only the identity mapping and ‘flipping the deck upside down’). I doubt anybody would restricting the possible permutations that much stil call shuffling.
The space is continuous even if we can only measure down to the Planck length.
I'm only a mathematician, no physicist. But I think to remember, that the concept of a continuous physical space becomes quite muddled at this scale.
If you don't want to assume continuity, then you get it back by rephrasing your theorems as "within a margin of error equal to the distance between discrete objects".
(I am not a professional mathematician, I might also be wrong.)
And of course, some functions/transformations are not continuous, and those may not have such an "axis of origin" at all.
The fact that the underlying space is contractible is very important here.